LMFDB - Embedded newform 1849.4.a.j.1.29

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,4,Mod(1,1849)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1849.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1849.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$109.094531601$
Analytic rank:	$1$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.29
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.02286 q^{2} -1.11016 q^{3} -6.95376 q^{4} -17.8254 q^{5} -1.13554 q^{6} -19.7420 q^{7} -15.2956 q^{8} -25.7675 q^{9} +O(q^{10})$	\(q+1.02286 q^{2} -1.11016 q^{3} -6.95376 q^{4} -17.8254 q^{5} -1.13554 q^{6} -19.7420 q^{7} -15.2956 q^{8} -25.7675 q^{9} -18.2329 q^{10} +22.7928 q^{11} +7.71979 q^{12} +19.7572 q^{13} -20.1933 q^{14} +19.7891 q^{15} +39.9848 q^{16} -101.707 q^{17} -26.3566 q^{18} +71.2415 q^{19} +123.954 q^{20} +21.9168 q^{21} +23.3139 q^{22} +39.5390 q^{23} +16.9806 q^{24} +192.745 q^{25} +20.2089 q^{26} +58.5805 q^{27} +137.281 q^{28} +100.371 q^{29} +20.2414 q^{30} +188.118 q^{31} +163.264 q^{32} -25.3037 q^{33} -104.032 q^{34} +351.910 q^{35} +179.181 q^{36} +431.553 q^{37} +72.8700 q^{38} -21.9337 q^{39} +272.650 q^{40} -388.086 q^{41} +22.4178 q^{42} -158.496 q^{44} +459.317 q^{45} +40.4428 q^{46} -444.898 q^{47} -44.3896 q^{48} +46.7482 q^{49} +197.151 q^{50} +112.911 q^{51} -137.387 q^{52} -561.950 q^{53} +59.9196 q^{54} -406.292 q^{55} +301.966 q^{56} -79.0895 q^{57} +102.665 q^{58} -582.977 q^{59} -137.608 q^{60} +490.093 q^{61} +192.418 q^{62} +508.704 q^{63} -152.883 q^{64} -352.181 q^{65} -25.8821 q^{66} +638.092 q^{67} +707.245 q^{68} -43.8946 q^{69} +359.955 q^{70} +37.3436 q^{71} +394.130 q^{72} -504.541 q^{73} +441.418 q^{74} -213.978 q^{75} -495.396 q^{76} -449.977 q^{77} -22.4351 q^{78} +690.289 q^{79} -712.746 q^{80} +630.690 q^{81} -396.958 q^{82} -112.701 q^{83} -152.404 q^{84} +1812.97 q^{85} -111.428 q^{87} -348.630 q^{88} +772.926 q^{89} +469.817 q^{90} -390.048 q^{91} -274.945 q^{92} -208.841 q^{93} -455.068 q^{94} -1269.91 q^{95} -181.249 q^{96} -403.914 q^{97} +47.8168 q^{98} -587.315 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9}+O(q^{10}) $ 50 * q + 10 * q^2 + 2 * q^3 + 186 * q^4 - 8 * q^5 - 51 * q^6 + 6 * q^7 + 138 * q^8 + 360 * q^9	\( 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9} - 137 q^{10} - 252 q^{11} + 48 q^{12} - 192 q^{13} - 272 q^{14} - 314 q^{15} + 542 q^{16} - 236 q^{17} + 386 q^{18} - 12 q^{19} - 108 q^{20} - 408 q^{21} - 1235 q^{22} - 630 q^{23} - 613 q^{24} + 1098 q^{25} - 1493 q^{26} - 10 q^{27} + 242 q^{28} - 208 q^{29} + 48 q^{30} - 932 q^{31} + 1124 q^{32} - 254 q^{33} - 765 q^{34} - 1452 q^{35} + 747 q^{36} + 90 q^{37} - 1213 q^{38} + 1610 q^{39} - 1693 q^{40} - 1354 q^{41} + 16 q^{42} - 2704 q^{44} - 4508 q^{45} - 233 q^{46} - 3484 q^{47} + 376 q^{48} + 1324 q^{49} + 408 q^{50} - 4054 q^{51} - 2176 q^{52} - 726 q^{53} - 6497 q^{54} + 3288 q^{55} - 7097 q^{56} - 870 q^{57} + 275 q^{58} - 4370 q^{59} - 3891 q^{60} - 1172 q^{61} + 1546 q^{62} + 3686 q^{63} + 606 q^{64} - 2610 q^{65} - 4697 q^{66} - 344 q^{67} - 3221 q^{68} - 136 q^{69} + 1310 q^{70} - 162 q^{71} + 5814 q^{72} - 746 q^{73} - 4332 q^{74} - 236 q^{75} - 1338 q^{76} - 2024 q^{77} - 2782 q^{78} - 2656 q^{79} + 5713 q^{80} - 86 q^{81} + 4168 q^{82} - 3514 q^{83} - 4269 q^{84} + 7558 q^{85} - 10278 q^{87} - 11692 q^{88} - 2640 q^{89} - 8286 q^{90} + 5946 q^{91} - 4271 q^{92} + 2 q^{93} - 9062 q^{94} - 12140 q^{95} - 700 q^{96} - 3864 q^{97} + 2826 q^{98} - 8174 q^{99}+O(q^{100}) \) 50 * q + 10 * q^2 + 2 * q^3 + 186 * q^4 - 8 * q^5 - 51 * q^6 + 6 * q^7 + 138 * q^8 + 360 * q^9 - 137 * q^10 - 252 * q^11 + 48 * q^12 - 192 * q^13 - 272 * q^14 - 314 * q^15 + 542 * q^16 - 236 * q^17 + 386 * q^18 - 12 * q^19 - 108 * q^20 - 408 * q^21 - 1235 * q^22 - 630 * q^23 - 613 * q^24 + 1098 * q^25 - 1493 * q^26 - 10 * q^27 + 242 * q^28 - 208 * q^29 + 48 * q^30 - 932 * q^31 + 1124 * q^32 - 254 * q^33 - 765 * q^34 - 1452 * q^35 + 747 * q^36 + 90 * q^37 - 1213 * q^38 + 1610 * q^39 - 1693 * q^40 - 1354 * q^41 + 16 * q^42 - 2704 * q^44 - 4508 * q^45 - 233 * q^46 - 3484 * q^47 + 376 * q^48 + 1324 * q^49 + 408 * q^50 - 4054 * q^51 - 2176 * q^52 - 726 * q^53 - 6497 * q^54 + 3288 * q^55 - 7097 * q^56 - 870 * q^57 + 275 * q^58 - 4370 * q^59 - 3891 * q^60 - 1172 * q^61 + 1546 * q^62 + 3686 * q^63 + 606 * q^64 - 2610 * q^65 - 4697 * q^66 - 344 * q^67 - 3221 * q^68 - 136 * q^69 + 1310 * q^70 - 162 * q^71 + 5814 * q^72 - 746 * q^73 - 4332 * q^74 - 236 * q^75 - 1338 * q^76 - 2024 * q^77 - 2782 * q^78 - 2656 * q^79 + 5713 * q^80 - 86 * q^81 + 4168 * q^82 - 3514 * q^83 - 4269 * q^84 + 7558 * q^85 - 10278 * q^87 - 11692 * q^88 - 2640 * q^89 - 8286 * q^90 + 5946 * q^91 - 4271 * q^92 + 2 * q^93 - 9062 * q^94 - 12140 * q^95 - 700 * q^96 - 3864 * q^97 + 2826 * q^98 - 8174 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.02286	0.361635	0.180818	−	0.983517i	$-0.442126\pi$
$2$	1.02286	0.361635	0.180818	+	0.983517i	$0.442126\pi$
$3$	−1.11016	−0.213651	−0.106825	−	0.994278i	$-0.534069\pi$
$3$	−1.11016	−0.213651	−0.106825	+	0.994278i	$0.534069\pi$
$4$	−6.95376	−0.869220
$5$	−17.8254	−1.59435	−0.797177	−	0.603746i	$-0.793674\pi$
$5$	−17.8254	−1.59435	−0.797177	+	0.603746i	$0.793674\pi$
$6$	−1.13554	−0.0772636
$7$	−19.7420	−1.06597	−0.532985	−	0.846125i	$-0.678930\pi$
$7$	−19.7420	−1.06597	−0.532985	+	0.846125i	$0.678930\pi$
$8$	−15.2956	−0.675976
$9$	−25.7675	−0.954353
$10$	−18.2329	−0.576575
$11$	22.7928	0.624754	0.312377	−	0.949958i	$-0.398875\pi$
$11$	22.7928	0.624754	0.312377	+	0.949958i	$0.398875\pi$
$12$	7.71979	0.185709
$13$	19.7572	0.421513	0.210757	−	0.977539i	$-0.432407\pi$
$13$	19.7572	0.421513	0.210757	+	0.977539i	$0.432407\pi$
$14$	−20.1933	−0.385493
$15$	19.7891	0.340634
$16$	39.9848	0.624763
$17$	−101.707	−1.45103	−0.725515	−	0.688206i	$-0.758399\pi$
$17$	−101.707	−1.45103	−0.725515	+	0.688206i	$0.758399\pi$
$18$	−26.3566	−0.345128
$19$	71.2415	0.860206	0.430103	−	0.902780i	$-0.358477\pi$
$19$	71.2415	0.860206	0.430103	+	0.902780i	$0.358477\pi$
$20$	123.954	1.38584
$21$	21.9168	0.227745
$22$	23.3139	0.225933
$23$	39.5390	0.358454	0.179227	−	0.983808i	$-0.442640\pi$
$23$	39.5390	0.358454	0.179227	+	0.983808i	$0.442640\pi$
$24$	16.9806	0.144423
$25$	192.745	1.54196
$26$	20.2089	0.152434
$27$	58.5805	0.417549
$28$	137.281	0.926562
$29$	100.371	0.642703	0.321351	−	0.946960i	$-0.395863\pi$
$29$	100.371	0.642703	0.321351	+	0.946960i	$0.395863\pi$
$30$	20.2414	0.123186
$31$	188.118	1.08990	0.544951	−	0.838468i	$-0.316548\pi$
$31$	188.118	1.08990	0.544951	+	0.838468i	$0.316548\pi$
$32$	163.264	0.901913
$33$	−25.3037	−0.133479
$34$	−104.032	−0.524744
$35$	351.910	1.69953
$36$	179.181	0.829543
$37$	431.553	1.91748	0.958741	−	0.284282i	$-0.0917551\pi$
$37$	431.553	1.91748	0.958741	+	0.284282i	$0.0917551\pi$
$38$	72.8700	0.311081
$39$	−21.9337	−0.0900565
$40$	272.650	1.07775
$41$	−388.086	−1.47827	−0.739133	−	0.673560i	$-0.764764\pi$
$41$	−388.086	−1.47827	−0.739133	+	0.673560i	$0.764764\pi$
$42$	22.4178	0.0823607
$43$	0	0
$43$	0	0
$44$	−158.496	−0.543049
$45$	459.317	1.52158
$46$	40.4428	0.129630
$47$	−444.898	−1.38075	−0.690373	−	0.723453i	$-0.742554\pi$
$47$	−444.898	−1.38075	−0.690373	+	0.723453i	$0.742554\pi$
$48$	−44.3896	−0.133481
$49$	46.7482	0.136292
$50$	197.151	0.557629
$51$	112.911	0.310014
$52$	−137.387	−0.366388
$53$	−561.950	−1.45641	−0.728206	−	0.685359i	$-0.759645\pi$
$53$	−561.950	−1.45641	−0.728206	+	0.685359i	$0.759645\pi$
$54$	59.9196	0.151000
$55$	−406.292	−0.996079
$56$	301.966	0.720570
$57$	−79.0895	−0.183783
$58$	102.665	0.232424
$59$	−582.977	−1.28639	−0.643196	−	0.765702i	$-0.722392\pi$
$59$	−582.977	−1.28639	−0.643196	+	0.765702i	$0.722392\pi$
$60$	−137.608	−0.296086
$61$	490.093	1.02869	0.514344	−	0.857584i	$-0.328036\pi$
$61$	490.093	1.02869	0.514344	+	0.857584i	$0.328036\pi$
$62$	192.418	0.394147
$63$	508.704	1.01731
$64$	−152.883	−0.298599
$65$	−352.181	−0.672041
$66$	−25.8821	−0.0482708
$67$	638.092	1.16351	0.581756	−	0.813363i	$-0.302366\pi$
$67$	638.092	1.16351	0.581756	+	0.813363i	$0.302366\pi$
$68$	707.245	1.26126
$69$	−43.8946	−0.0765839
$70$	359.955	0.614611
$71$	37.3436	0.0624207	0.0312103	−	0.999513i	$-0.490064\pi$
$71$	37.3436	0.0624207	0.0312103	+	0.999513i	$0.490064\pi$
$72$	394.130	0.645120
$73$	−504.541	−0.808933	−0.404466	−	0.914553i	$-0.632543\pi$
$73$	−504.541	−0.808933	−0.404466	+	0.914553i	$0.632543\pi$
$74$	441.418	0.693429
$75$	−213.978	−0.329441
$76$	−495.396	−0.747708
$77$	−449.977	−0.665969
$78$	−22.4351	−0.0325676
$79$	690.289	0.983084	0.491542	−	0.870854i	$-0.336433\pi$
$79$	690.289	0.983084	0.491542	+	0.870854i	$0.336433\pi$
$80$	−712.746	−0.996093
$81$	630.690	0.865144
$82$	−396.958	−0.534593
$83$	−112.701	−0.149043	−0.0745216	−	0.997219i	$-0.523743\pi$
$83$	−112.701	−0.149043	−0.0745216	+	0.997219i	$0.523743\pi$
$84$	−152.404	−0.197961
$85$	1812.97	2.31346
$86$	0	0
$87$	−111.428	−0.137314
$88$	−348.630	−0.422319
$89$	772.926	0.920562	0.460281	−	0.887773i	$-0.347749\pi$
$89$	772.926	0.920562	0.460281	+	0.887773i	$0.347749\pi$
$90$	469.817	0.550256
$91$	−390.048	−0.449321
$92$	−274.945	−0.311576
$93$	−208.841	−0.232858
$94$	−455.068	−0.499327
$95$	−1269.91	−1.37147
$96$	−181.249	−0.192694
$97$	−403.914	−0.422797	−0.211398	−	0.977400i	$-0.567802\pi$
$97$	−403.914	−0.422797	−0.211398	+	0.977400i	$0.567802\pi$
$98$	47.8168	0.0492881
$99$	−587.315	−0.596236
$100$	−1340.30	−1.34030
$101$	−550.505	−0.542350	−0.271175	−	0.962530i	$-0.587412\pi$
$101$	−550.505	−0.542350	−0.271175	+	0.962530i	$0.587412\pi$
$102$	115.492	0.112112
$103$	1818.49	1.73962	0.869812	−	0.493384i	$-0.164240\pi$
$103$	1818.49	1.73962	0.869812	+	0.493384i	$0.164240\pi$
$104$	−302.199	−0.284933
$105$	−390.677	−0.363106
$106$	−574.796	−0.526690
$107$	78.4386	0.0708686	0.0354343	−	0.999372i	$-0.488719\pi$
$107$	78.4386	0.0708686	0.0354343	+	0.999372i	$0.488719\pi$
$108$	−407.354	−0.362942
$109$	−157.371	−0.138288	−0.0691440	−	0.997607i	$-0.522027\pi$
$109$	−157.371	−0.138288	−0.0691440	+	0.997607i	$0.522027\pi$
$110$	−415.579	−0.360218
$111$	−479.093	−0.409671
$112$	−789.382	−0.665978
$113$	348.983	0.290527	0.145263	−	0.989393i	$-0.453597\pi$
$113$	348.983	0.290527	0.145263	+	0.989393i	$0.453597\pi$
$114$	−80.8974	−0.0664626
$115$	−704.799	−0.571503
$116$	−697.954	−0.558650
$117$	−509.096	−0.402273
$118$	−596.304	−0.465205
$119$	2007.90	1.54676
$120$	−302.686	−0.230261
$121$	−811.487	−0.609682
$122$	501.296	0.372010
$123$	430.838	0.315832
$124$	−1308.12	−0.947364
$125$	−1207.59	−0.864081
$126$	520.333	0.367896
$127$	1286.15	0.898643	0.449322	−	0.893370i	$-0.351666\pi$
$127$	1286.15	0.898643	0.449322	+	0.893370i	$0.351666\pi$
$128$	−1462.49	−1.00990
$129$	0	0
$130$	−360.232	−0.243034
$131$	−2694.05	−1.79680	−0.898399	−	0.439181i	$-0.855269\pi$
$131$	−2694.05	−1.79680	−0.898399	+	0.439181i	$0.855269\pi$
$132$	175.956	0.116023
$133$	−1406.45	−0.916954
$134$	652.678	0.420767
$135$	−1044.22	−0.665720
$136$	1555.67	0.980862
$137$	−21.0377	−0.0131195	−0.00655976	−	0.999978i	$-0.502088\pi$
$137$	−21.0377	−0.0131195	−0.00655976	+	0.999978i	$0.502088\pi$
$138$	−44.8980	−0.0276955
$139$	3054.31	1.86376	0.931882	−	0.362761i	$-0.118166\pi$
$139$	3054.31	1.86376	0.931882	+	0.362761i	$0.118166\pi$
$140$	−2447.10	−1.47727
$141$	493.909	0.294997
$142$	38.1973	0.0225735
$143$	450.323	0.263342
$144$	−1030.31	−0.596245
$145$	−1789.15	−1.02470
$146$	−516.075	−0.292539
$147$	−51.8980	−0.0291189
$148$	−3000.91	−1.66671
$149$	1300.51	0.715047	0.357524	−	0.933904i	$-0.383621\pi$
$149$	1300.51	0.715047	0.357524	+	0.933904i	$0.383621\pi$
$150$	−218.870	−0.119138
$151$	1123.62	0.605556	0.302778	−	0.953061i	$-0.402086\pi$
$151$	1123.62	0.605556	0.302778	+	0.953061i	$0.402086\pi$
$152$	−1089.68	−0.581479
$153$	2620.73	1.38480
$154$	−460.263	−0.240838
$155$	−3353.28	−1.73769
$156$	152.522	0.0782789
$157$	−2420.82	−1.23059	−0.615294	−	0.788298i	$-0.710963\pi$
$157$	−2420.82	−1.23059	−0.615294	+	0.788298i	$0.710963\pi$
$158$	706.069	0.355518
$159$	623.855	0.311163
$160$	−2910.24	−1.43797
$161$	−780.580	−0.382102
$162$	645.107	0.312867
$163$	2633.91	1.26567	0.632834	−	0.774288i	$-0.281892\pi$
$163$	2633.91	1.26567	0.632834	+	0.774288i	$0.281892\pi$
$164$	2698.66	1.28494
$165$	451.049	0.212813
$166$	−115.278	−0.0538993
$167$	−2477.45	−1.14797	−0.573984	−	0.818866i	$-0.694603\pi$
$167$	−2477.45	−1.14797	−0.573984	+	0.818866i	$0.694603\pi$
$168$	−335.231	−0.153950
$169$	−1806.65	−0.822327
$170$	1854.41	0.836628
$171$	−1835.72	−0.820941
$172$	0	0
$173$	1641.92	0.721575	0.360788	−	0.932648i	$-0.382508\pi$
$173$	1641.92	0.721575	0.360788	+	0.932648i	$0.382508\pi$
$174$	−113.975	−0.0496575
$175$	−3805.19	−1.64369
$176$	911.367	0.390323
$177$	647.198	0.274838
$178$	790.595	0.332908
$179$	−197.274	−0.0823742	−0.0411871	−	0.999151i	$-0.513114\pi$
$179$	−197.274	−0.0823742	−0.0411871	+	0.999151i	$0.513114\pi$
$180$	−3193.98	−1.32258
$181$	−2120.28	−0.870714	−0.435357	−	0.900258i	$-0.643378\pi$
$181$	−2120.28	−0.870714	−0.435357	+	0.900258i	$0.643378\pi$
$182$	−398.965	−0.162490
$183$	−544.081	−0.219780
$184$	−604.772	−0.242307
$185$	−7692.61	−3.05714
$186$	−213.615	−0.0842097
$187$	−2318.19	−0.906538
$188$	3093.71	1.20017
$189$	−1156.50	−0.445094
$190$	−1298.94	−0.495973
$191$	3247.98	1.23045	0.615223	−	0.788353i	$-0.289066\pi$
$191$	3247.98	1.23045	0.615223	+	0.788353i	$0.289066\pi$
$192$	169.724	0.0637959
$193$	4635.04	1.72869	0.864345	−	0.502899i	$-0.167733\pi$
$193$	4635.04	1.72869	0.864345	+	0.502899i	$0.167733\pi$
$194$	−413.147	−0.152898
$195$	390.978	0.143582
$196$	−325.076	−0.118468
$197$	−2130.99	−0.770695	−0.385348	−	0.922772i	$-0.625918\pi$
$197$	−2130.99	−0.770695	−0.385348	+	0.922772i	$0.625918\pi$
$198$	−600.741	−0.215620
$199$	3441.83	1.22605	0.613027	−	0.790062i	$-0.289952\pi$
$199$	3441.83	1.22605	0.613027	+	0.790062i	$0.289952\pi$
$200$	−2948.16	−1.04233
$201$	−708.385	−0.248585
$202$	−563.090	−0.196133
$203$	−1981.52	−0.685102
$204$	−785.155	−0.269470
$205$	6917.80	2.35688
$206$	1860.06	0.629110
$207$	−1018.82	−0.342092
$208$	789.990	0.263346
$209$	1623.80	0.537417
$210$	−399.607	−0.131312
$211$	−1835.67	−0.598922	−0.299461	−	0.954109i	$-0.596807\pi$
$211$	−1835.67	−0.598922	−0.299461	+	0.954109i	$0.596807\pi$
$212$	3907.67	1.26594
$213$	−41.4574	−0.0133362
$214$	80.2317	0.0256286
$215$	0	0
$216$	−896.023	−0.282253
$217$	−3713.83	−1.16180
$218$	−160.968	−0.0500098
$219$	560.122	0.172829
$220$	2825.25	0.865812
$221$	−2009.45	−0.611629
$222$	−490.045	−0.148152
$223$	2382.48	0.715438	0.357719	−	0.933829i	$-0.383555\pi$
$223$	2382.48	0.715438	0.357719	+	0.933829i	$0.383555\pi$
$224$	−3223.16	−0.961412
$225$	−4966.58	−1.47158
$226$	356.960	0.105065
$227$	1427.19	0.417294	0.208647	−	0.977991i	$-0.433094\pi$
$227$	1427.19	0.417294	0.208647	+	0.977991i	$0.433094\pi$
$228$	549.969	0.159748
$229$	−3610.07	−1.04175	−0.520874	−	0.853634i	$-0.674394\pi$
$229$	−3610.07	−1.04175	−0.520874	+	0.853634i	$0.674394\pi$
$230$	−720.910	−0.206676
$231$	499.547	0.142285
$232$	−1535.23	−0.434452
$233$	−844.802	−0.237532	−0.118766	−	0.992922i	$-0.537894\pi$
$233$	−844.802	−0.237532	−0.118766	+	0.992922i	$0.537894\pi$
$234$	−520.733	−0.145476
$235$	7930.50	2.20140
$236$	4053.88	1.11816
$237$	−766.332	−0.210036
$238$	2053.80	0.559362
$239$	3678.94	0.995693	0.497846	−	0.867265i	$-0.334124\pi$
$239$	3678.94	0.995693	0.497846	+	0.867265i	$0.334124\pi$
$240$	791.263	0.212816
$241$	−2907.74	−0.777194	−0.388597	−	0.921408i	$-0.627040\pi$
$241$	−2907.74	−0.777194	−0.388597	+	0.921408i	$0.627040\pi$
$242$	−830.037	−0.220483
$243$	−2281.84	−0.602387
$244$	−3407.98	−0.894155
$245$	−833.306	−0.217298
$246$	440.687	0.114216
$247$	1407.54	0.362588
$248$	−2877.37	−0.736747
$249$	125.117	0.0318432
$250$	−1235.20	−0.312482
$251$	7305.43	1.83711	0.918555	−	0.395294i	$-0.129357\pi$
$251$	7305.43	1.83711	0.918555	+	0.395294i	$0.129357\pi$
$252$	−3537.40	−0.884268
$253$	901.205	0.223946
$254$	1315.56	0.324981
$255$	−2012.68	−0.494271
$256$	−272.856	−0.0666153
$257$	1564.64	0.379765	0.189882	−	0.981807i	$-0.439189\pi$
$257$	1564.64	0.379765	0.189882	+	0.981807i	$0.439189\pi$
$258$	0	0
$259$	−8519.73	−2.04398
$260$	2448.98	0.584151
$261$	−2586.31	−0.613365
$262$	−2755.64	−0.649786
$263$	1714.84	0.402060	0.201030	−	0.979585i	$-0.435571\pi$
$263$	1714.84	0.402060	0.201030	+	0.979585i	$0.435571\pi$
$264$	387.035	0.0902287
$265$	10017.0	2.32203
$266$	−1438.60	−0.331603
$267$	−858.072	−0.196678
$268$	−4437.14	−1.01135
$269$	−5451.01	−1.23552	−0.617758	−	0.786368i	$-0.711959\pi$
$269$	−5451.01	−1.23552	−0.617758	+	0.786368i	$0.711959\pi$
$270$	−1068.09	−0.240748
$271$	−1315.31	−0.294832	−0.147416	−	0.989075i	$-0.547096\pi$
$271$	−1315.31	−0.294832	−0.147416	+	0.989075i	$0.547096\pi$
$272$	−4066.73	−0.906550
$273$	433.016	0.0959976
$274$	−21.5186	−0.00474448
$275$	4393.21	0.963348
$276$	305.233	0.0665683
$277$	2640.70	0.572795	0.286398	−	0.958111i	$-0.407542\pi$
$277$	2640.70	0.572795	0.286398	+	0.958111i	$0.407542\pi$
$278$	3124.13	0.674003
$279$	−4847.33	−1.04015
$280$	−5382.67	−1.14884
$281$	−4294.12	−0.911621	−0.455811	−	0.890077i	$-0.650651\pi$
$281$	−4294.12	−0.911621	−0.455811	+	0.890077i	$0.650651\pi$
$282$	505.199	0.106681
$283$	−3322.87	−0.697965	−0.348983	−	0.937129i	$-0.613473\pi$
$283$	−3322.87	−0.697965	−0.348983	+	0.937129i	$0.613473\pi$
$284$	−259.678	−0.0542573
$285$	1409.80	0.293016
$286$	460.618	0.0952339
$287$	7661.62	1.57579
$288$	−4206.90	−0.860743
$289$	5431.28	1.10549
$290$	−1830.05	−0.370566
$291$	448.410	0.0903307
$292$	3508.46	0.703141
$293$	−190.708	−0.0380249	−0.0190125	−	0.999819i	$-0.506052\pi$
$293$	−190.708	−0.0380249	−0.0190125	+	0.999819i	$0.506052\pi$
$294$	−53.0844	−0.0105304
$295$	10391.8	2.05096
$296$	−6600.86	−1.29617
$297$	1335.21	0.260865
$298$	1330.24	0.258586
$299$	781.181	0.151093
$300$	1487.95	0.286357
$301$	0	0
$302$	1149.31	0.218990
$303$	611.149	0.115873
$304$	2848.58	0.537425
$305$	−8736.10	−1.64009
$306$	2680.64	0.500792
$307$	4341.31	0.807073	0.403536	−	0.914964i	$-0.367781\pi$
$307$	4341.31	0.807073	0.403536	+	0.914964i	$0.367781\pi$
$308$	3129.03	0.578874
$309$	−2018.82	−0.371672
$310$	−3429.93	−0.628410
$311$	−6092.04	−1.11077	−0.555383	−	0.831595i	$-0.687428\pi$
$311$	−6092.04	−1.11077	−0.555383	+	0.831595i	$0.687428\pi$
$312$	335.489	0.0608761
$313$	−5612.18	−1.01348	−0.506740	−	0.862099i	$-0.669149\pi$
$313$	−5612.18	−1.01348	−0.506740	+	0.862099i	$0.669149\pi$
$314$	−2476.16	−0.445024
$315$	−9067.86	−1.62196
$316$	−4800.10	−0.854516
$317$	−3963.80	−0.702299	−0.351150	−	0.936319i	$-0.614209\pi$
$317$	−3963.80	−0.702299	−0.351150	+	0.936319i	$0.614209\pi$
$318$	638.116	0.112528
$319$	2287.73	0.401531
$320$	2725.20	0.476073
$321$	−87.0794	−0.0151411
$322$	−798.424	−0.138181
$323$	−7245.74	−1.24819
$324$	−4385.67	−0.752000
$325$	3808.12	0.649958
$326$	2694.12	0.457710
$327$	174.707	0.0295453
$328$	5936.01	0.999273
$329$	8783.20	1.47183
$330$	461.360	0.0769607
$331$	−9099.39	−1.51102	−0.755510	−	0.655137i	$-0.772611\pi$
$331$	−9099.39	−1.51102	−0.755510	+	0.655137i	$0.772611\pi$
$332$	783.698	0.129551
$333$	−11120.1	−1.82996
$334$	−2534.08	−0.415146
$335$	−11374.3	−1.85505
$336$	876.341	0.142287
$337$	−2484.89	−0.401663	−0.200832	−	0.979626i	$-0.564364\pi$
$337$	−2484.89	−0.401663	−0.200832	+	0.979626i	$0.564364\pi$
$338$	−1847.95	−0.297382
$339$	−387.427	−0.0620712
$340$	−12606.9	−2.01090
$341$	4287.73	0.680920
$342$	−1877.68	−0.296881
$343$	5848.62	0.920687
$344$	0	0
$345$	782.440	0.122102
$346$	1679.45	0.260947
$347$	9452.41	1.46234	0.731170	−	0.682196i	$-0.238975\pi$
$347$	9452.41	1.46234	0.731170	+	0.682196i	$0.238975\pi$
$348$	774.841	0.119356
$349$	−2929.71	−0.449352	−0.224676	−	0.974434i	$-0.572132\pi$
$349$	−2929.71	−0.449352	−0.224676	+	0.974434i	$0.572132\pi$
$350$	−3892.17	−0.594415
$351$	1157.39	0.176002
$352$	3721.24	0.563474
$353$	1497.90	0.225851	0.112926	−	0.993603i	$-0.463978\pi$
$353$	1497.90	0.225851	0.112926	+	0.993603i	$0.463978\pi$
$354$	661.993	0.0993913
$355$	−665.665	−0.0995207
$356$	−5374.74	−0.800170
$357$	−2229.09	−0.330465
$358$	−201.784	−0.0297894
$359$	11573.3	1.70144	0.850721	−	0.525618i	$-0.176166\pi$
$359$	11573.3	1.70144	0.850721	+	0.525618i	$0.176166\pi$
$360$	−7025.53	−1.02855
$361$	−1783.65	−0.260045
$362$	−2168.75	−0.314881
$363$	900.881	0.130259
$364$	2712.30	0.390558
$365$	8993.66	1.28973
$366$	−556.519	−0.0794801
$367$	−3815.58	−0.542702	−0.271351	−	0.962480i	$-0.587470\pi$
$367$	−3815.58	−0.542702	−0.271351	+	0.962480i	$0.587470\pi$
$368$	1580.96	0.223949
$369$	10000.0	1.41079
$370$	−7868.46	−1.10557
$371$	11094.0	1.55249
$372$	1452.23	0.202405
$373$	1969.77	0.273434	0.136717	−	0.990610i	$-0.456345\pi$
$373$	1969.77	0.273434	0.136717	+	0.990610i	$0.456345\pi$
$374$	−2371.18	−0.327836
$375$	1340.62	0.184611
$376$	6804.98	0.933352
$377$	1983.05	0.270908
$378$	−1182.93	−0.160962
$379$	1267.89	0.171839	0.0859194	−	0.996302i	$-0.472617\pi$
$379$	1267.89	0.171839	0.0859194	+	0.996302i	$0.472617\pi$
$380$	8830.64	1.19211
$381$	−1427.84	−0.191996
$382$	3322.22	0.444973
$383$	−2523.67	−0.336694	−0.168347	−	0.985728i	$-0.553843\pi$
$383$	−2523.67	−0.336694	−0.168347	+	0.985728i	$0.553843\pi$
$384$	1623.60	0.215765
$385$	8021.03	1.06179
$386$	4740.99	0.625156
$387$	0	0
$388$	2808.72	0.367503
$389$	−12576.9	−1.63926	−0.819631	−	0.572892i	$-0.805821\pi$
$389$	−12576.9	−1.63926	−0.819631	+	0.572892i	$0.805821\pi$
$390$	399.915	0.0519243
$391$	−4021.38	−0.520128
$392$	−715.041	−0.0921302
$393$	2990.83	0.383887
$394$	−2179.71	−0.278711
$395$	−12304.7	−1.56738
$396$	4084.05	0.518260
$397$	−6253.88	−0.790613	−0.395306	−	0.918549i	$-0.629362\pi$
$397$	−6253.88	−0.790613	−0.395306	+	0.918549i	$0.629362\pi$
$398$	3520.51	0.443384
$399$	1561.39	0.195908
$400$	7706.89	0.963361
$401$	1382.58	0.172177	0.0860883	−	0.996288i	$-0.472563\pi$
$401$	1382.58	0.172177	0.0860883	+	0.996288i	$0.472563\pi$
$402$	−724.578	−0.0898972
$403$	3716.69	0.459408
$404$	3828.08	0.471421
$405$	−11242.3	−1.37935
$406$	−2026.82	−0.247757
$407$	9836.31	1.19795
$408$	−1727.04	−0.209562
$409$	13695.7	1.65577	0.827885	−	0.560899i	$-0.189544\pi$
$409$	13695.7	1.65577	0.827885	+	0.560899i	$0.189544\pi$
$410$	7075.94	0.852331
$411$	23.3552	0.00280299
$412$	−12645.3	−1.51212
$413$	11509.2	1.37126
$414$	−1042.11	−0.123713
$415$	2008.95	0.237628
$416$	3225.64	0.380168
$417$	−3390.78	−0.398194
$418$	1660.91	0.194349
$419$	−10610.7	−1.23715	−0.618573	−	0.785727i	$-0.712289\pi$
$419$	−10610.7	−1.23715	−0.618573	+	0.785727i	$0.712289\pi$
$420$	2716.67	0.315619
$421$	2110.80	0.244357	0.122178	−	0.992508i	$-0.461012\pi$
$421$	2110.80	0.244357	0.122178	+	0.992508i	$0.461012\pi$
$422$	−1877.63	−0.216592
$423$	11463.9	1.31772
$424$	8595.36	0.984499
$425$	−19603.5	−2.23744
$426$	−42.4051	−0.00482285
$427$	−9675.43	−1.09655
$428$	−545.443	−0.0616004
$429$	−499.931	−0.0562632
$430$	0	0
$431$	12954.3	1.44777	0.723883	−	0.689922i	$-0.242355\pi$
$431$	12954.3	1.44777	0.723883	+	0.689922i	$0.242355\pi$
$432$	2342.33	0.260869
$433$	−1171.99	−0.130075	−0.0650374	−	0.997883i	$-0.520717\pi$
$433$	−1171.99	−0.130075	−0.0650374	+	0.997883i	$0.520717\pi$
$434$	−3798.72	−0.420149
$435$	1986.24	0.218927
$436$	1094.32	0.120203
$437$	2816.82	0.308345
$438$	572.926	0.0625011
$439$	−16402.2	−1.78322	−0.891609	−	0.452807i	$-0.850423\pi$
$439$	−16402.2	−1.78322	−0.891609	+	0.452807i	$0.850423\pi$
$440$	6214.47	0.673326
$441$	−1204.59	−0.130071
$442$	−2055.38	−0.221187
$443$	10078.9	1.08096	0.540479	−	0.841358i	$-0.318243\pi$
$443$	10078.9	1.08096	0.540479	+	0.841358i	$0.318243\pi$
$444$	3331.50	0.356094
$445$	−13777.7	−1.46770
$446$	2436.94	0.258728
$447$	−1443.78	−0.152770
$448$	3018.22	0.318298
$449$	−6527.47	−0.686081	−0.343040	−	0.939321i	$-0.611457\pi$
$449$	−6527.47	−0.686081	−0.343040	+	0.939321i	$0.611457\pi$
$450$	−5080.11	−0.532175
$451$	−8845.59	−0.923553
$452$	−2426.74	−0.252532
$453$	−1247.40	−0.129377
$454$	1459.81	0.150908
$455$	6952.77	0.716376
$456$	1209.72	0.124233
$457$	−5400.42	−0.552782	−0.276391	−	0.961045i	$-0.589138\pi$
$457$	−5400.42	−0.552782	−0.276391	+	0.961045i	$0.589138\pi$
$458$	−3692.60	−0.376733
$459$	−5958.03	−0.605876
$460$	4901.00	0.496762
$461$	−6304.64	−0.636956	−0.318478	−	0.947930i	$-0.603172\pi$
$461$	−6304.64	−0.636956	−0.318478	+	0.947930i	$0.603172\pi$
$462$	510.966	0.0514552
$463$	19243.6	1.93158	0.965792	−	0.259317i	$-0.0834974\pi$
$463$	19243.6	1.93158	0.965792	+	0.259317i	$0.0834974\pi$
$464$	4013.30	0.401537
$465$	3722.68	0.371258
$466$	−864.114	−0.0858998
$467$	−2714.13	−0.268940	−0.134470	−	0.990918i	$-0.542933\pi$
$467$	−2714.13	−0.268940	−0.134470	+	0.990918i	$0.542933\pi$
$468$	3540.13	0.349663
$469$	−12597.2	−1.24027
$470$	8111.78	0.796104
$471$	2687.50	0.262916
$472$	8916.98	0.869570
$473$	0	0
$474$	−783.850	−0.0759566
$475$	13731.5	1.32641
$476$	−13962.5	−1.34447
$477$	14480.1	1.38993
$478$	3763.03	0.360078
$479$	−11210.3	−1.06933	−0.534666	−	0.845063i	$-0.679563\pi$
$479$	−11210.3	−1.06933	−0.534666	+	0.845063i	$0.679563\pi$
$480$	3230.84	0.307223
$481$	8526.29	0.808244
$482$	−2974.21	−0.281061
$483$	866.570	0.0816362
$484$	5642.88	0.529948
$485$	7199.94	0.674087
$486$	−2334.00	−0.217845
$487$	8701.07	0.809617	0.404808	−	0.914402i	$-0.367338\pi$
$487$	8701.07	0.809617	0.404808	+	0.914402i	$0.367338\pi$
$488$	−7496.26	−0.695368
$489$	−2924.06	−0.270410
$490$	−852.355	−0.0785826
$491$	−1624.22	−0.149288	−0.0746438	−	0.997210i	$-0.523782\pi$
$491$	−1624.22	−0.149288	−0.0746438	+	0.997210i	$0.523782\pi$
$492$	−2995.94	−0.274528
$493$	−10208.4	−0.932581
$494$	1439.71	0.131125
$495$	10469.1	0.950612
$496$	7521.85	0.680930
$497$	−737.239	−0.0665386
$498$	127.977	0.0115156
$499$	−14069.2	−1.26217	−0.631084	−	0.775714i	$-0.717390\pi$
$499$	−14069.2	−1.26217	−0.631084	+	0.775714i	$0.717390\pi$
$500$	8397.29	0.751076
$501$	2750.37	0.245264
$502$	7472.43	0.664364
$503$	11488.0	1.01834	0.509170	−	0.860666i	$-0.329952\pi$
$503$	11488.0	1.01834	0.509170	+	0.860666i	$0.329952\pi$
$504$	−7780.93	−0.687679
$505$	9812.98	0.864697
$506$	921.807	0.0809868
$507$	2005.67	0.175691
$508$	−8943.60	−0.781119
$509$	−9359.44	−0.815030	−0.407515	−	0.913199i	$-0.633605\pi$
$509$	−9359.44	−0.815030	−0.407515	+	0.913199i	$0.633605\pi$
$510$	−2058.69	−0.178746
$511$	9960.68	0.862298
$512$	11420.8	0.985806
$513$	4173.36	0.359178
$514$	1600.41	0.137336
$515$	−32415.4	−2.77358
$516$	0	0
$517$	−10140.5	−0.862627
$518$	−8714.49	−0.739175
$519$	−1822.79	−0.154165
$520$	5386.82	0.454284
$521$	−1027.52	−0.0864036	−0.0432018	−	0.999066i	$-0.513756\pi$
$521$	−1027.52	−0.0864036	−0.0432018	+	0.999066i	$0.513756\pi$
$522$	−2645.43	−0.221815
$523$	9348.97	0.781648	0.390824	−	0.920465i	$-0.372190\pi$
$523$	9348.97	0.781648	0.390824	+	0.920465i	$0.372190\pi$
$524$	18733.8	1.56181
$525$	4224.37	0.351175
$526$	1754.04	0.145399
$527$	−19132.9	−1.58148
$528$	−1011.76	−0.0833928
$529$	−10603.7	−0.871511
$530$	10246.0	0.839730
$531$	15021.9	1.22767
$532$	9780.13	0.797035
$533$	−7667.51	−0.623109
$534$	−877.687	−0.0711259
$535$	−1398.20	−0.112990
$536$	−9760.00	−0.786507
$537$	219.006	0.0175993
$538$	−5575.61	−0.446806
$539$	1065.52	0.0851491
$540$	7261.26	0.578657
$541$	−19965.3	−1.58664	−0.793322	−	0.608802i	$-0.791650\pi$
$541$	−19965.3	−1.58664	−0.793322	+	0.608802i	$0.791650\pi$
$542$	−1345.38	−0.106622
$543$	2353.85	0.186029
$544$	−16605.0	−1.30870
$545$	2805.20	0.220480
$546$	442.915	0.0347161
$547$	−15813.1	−1.23605	−0.618027	−	0.786157i	$-0.712068\pi$
$547$	−15813.1	−1.23605	−0.618027	+	0.786157i	$0.712068\pi$
$548$	146.291	0.0114037
$549$	−12628.5	−0.981731
$550$	4493.64	0.348381
$551$	7150.56	0.552857
$552$	671.394	0.0517689
$553$	−13627.7	−1.04794
$554$	2701.07	0.207143
$555$	8540.03	0.653160
$556$	−21238.9	−1.62002
$557$	−3373.07	−0.256592	−0.128296	−	0.991736i	$-0.540951\pi$
$557$	−3373.07	−0.256592	−0.128296	+	0.991736i	$0.540951\pi$
$558$	−4958.14	−0.376155
$559$	0	0
$560$	14071.1	1.06181
$561$	2573.56	0.193682
$562$	−4392.28	−0.329675
$563$	21680.8	1.62298	0.811489	−	0.584368i	$-0.198657\pi$
$563$	21680.8	1.62298	0.811489	+	0.584368i	$0.198657\pi$
$564$	−3434.52	−0.256417
$565$	−6220.76	−0.463203
$566$	−3398.83	−0.252409
$567$	−12451.1	−0.922218
$568$	−571.193	−0.0421949
$569$	3600.43	0.265268	0.132634	−	0.991165i	$-0.457656\pi$
$569$	3600.43	0.265268	0.132634	+	0.991165i	$0.457656\pi$
$570$	1442.03	0.105965
$571$	−15694.5	−1.15025	−0.575126	−	0.818065i	$-0.695047\pi$
$571$	−15694.5	−1.15025	−0.575126	+	0.818065i	$0.695047\pi$
$572$	−3131.44	−0.228902
$573$	−3605.78	−0.262886
$574$	7836.76	0.569861
$575$	7620.96	0.552723
$576$	3939.41	0.284969
$577$	−8951.52	−0.645852	−0.322926	−	0.946424i	$-0.604666\pi$
$577$	−8951.52	−0.645852	−0.322926	+	0.946424i	$0.604666\pi$
$578$	5555.43	0.399785
$579$	−5145.64	−0.369336
$580$	12441.3	0.890685
$581$	2224.96	0.158876
$582$	458.660	0.0326668
$583$	−12808.4	−0.909899
$584$	7717.26	0.546819
$585$	9074.84	0.641365
$586$	−195.068	−0.0137512
$587$	−4071.25	−0.286266	−0.143133	−	0.989703i	$-0.545718\pi$
$587$	−4071.25	−0.286266	−0.143133	+	0.989703i	$0.545718\pi$
$588$	360.886	0.0253107
$589$	13401.8	0.937540
$590$	10629.4	0.741701
$591$	2365.74	0.164659
$592$	17255.6	1.19797
$593$	7458.47	0.516497	0.258249	−	0.966079i	$-0.416855\pi$
$593$	7458.47	0.516497	0.258249	+	0.966079i	$0.416855\pi$
$594$	1365.74	0.0943382
$595$	−35791.7	−2.46608
$596$	−9043.44	−0.621533
$597$	−3820.98	−0.261947
$598$	799.039	0.0546407
$599$	3861.66	0.263411	0.131705	−	0.991289i	$-0.457955\pi$
$599$	3861.66	0.263411	0.131705	+	0.991289i	$0.457955\pi$
$600$	3272.93	0.222694
$601$	−3695.64	−0.250829	−0.125414	−	0.992104i	$-0.540026\pi$
$601$	−3695.64	−0.250829	−0.125414	+	0.992104i	$0.540026\pi$
$602$	0	0
$603$	−16442.1	−1.11040
$604$	−7813.38	−0.526361
$605$	14465.1	0.972049
$606$	625.120	0.0419039
$607$	−7398.42	−0.494716	−0.247358	−	0.968924i	$-0.579562\pi$
$607$	−7398.42	−0.494716	−0.247358	+	0.968924i	$0.579562\pi$
$608$	11631.1	0.775831
$609$	2199.81	0.146372
$610$	−8935.81	−0.593115
$611$	−8789.96	−0.582003
$612$	−18224.0	−1.20369
$613$	18359.2	1.20966	0.604828	−	0.796356i	$-0.293242\pi$
$613$	18359.2	1.20966	0.604828	+	0.796356i	$0.293242\pi$
$614$	4440.55	0.291866
$615$	−7679.87	−0.503548
$616$	6882.67	0.450179
$617$	−12505.8	−0.815986	−0.407993	−	0.912985i	$-0.633771\pi$
$617$	−12505.8	−0.815986	−0.407993	+	0.912985i	$0.633771\pi$
$618$	−2064.97	−0.134410
$619$	10523.0	0.683288	0.341644	−	0.939829i	$-0.389016\pi$
$619$	10523.0	0.683288	0.341644	+	0.939829i	$0.389016\pi$
$620$	23317.9	1.51043
$621$	2316.21	0.149672
$622$	−6231.31	−0.401692
$623$	−15259.1	−0.981291
$624$	−877.015	−0.0562640
$625$	−2567.38	−0.164313
$626$	−5740.47	−0.366510
$627$	−1802.67	−0.114820
$628$	16833.8	1.06965
$629$	−43891.8	−2.78233
$630$	−9275.15	−0.586557
$631$	−28065.1	−1.77061	−0.885306	−	0.465010i	$-0.846051\pi$
$631$	−28065.1	−1.77061	−0.885306	+	0.465010i	$0.846051\pi$
$632$	−10558.4	−0.664541
$633$	2037.89	0.127960
$634$	−4054.41	−0.253976
$635$	−22926.2	−1.43276
$636$	−4338.14	−0.270469
$637$	923.615	0.0574489
$638$	2340.03	0.145208
$639$	−962.253	−0.0595714
$640$	26069.4	1.61013
$641$	20560.3	1.26690	0.633450	−	0.773784i	$-0.281638\pi$
$641$	20560.3	1.26690	0.633450	+	0.773784i	$0.281638\pi$
$642$	−89.0700	−0.00547557
$643$	1245.51	0.0763889	0.0381944	−	0.999270i	$-0.487839\pi$
$643$	1245.51	0.0763889	0.0381944	+	0.999270i	$0.487839\pi$
$644$	5427.97	0.332130
$645$	0	0
$646$	−7411.38	−0.451388
$647$	11323.3	0.688047	0.344024	−	0.938961i	$-0.388210\pi$
$647$	11323.3	0.688047	0.344024	+	0.938961i	$0.388210\pi$
$648$	−9646.78	−0.584817
$649$	−13287.7	−0.803679
$650$	3895.17	0.235048
$651$	4122.95	0.248220
$652$	−18315.6	−1.10014
$653$	4387.92	0.262960	0.131480	−	0.991319i	$-0.458027\pi$
$653$	4387.92	0.262960	0.131480	+	0.991319i	$0.458027\pi$
$654$	178.701	0.0106846
$655$	48022.6	2.86473
$656$	−15517.6	−0.923566
$657$	13000.8	0.772008
$658$	8983.98	0.532267
$659$	−4067.28	−0.240423	−0.120211	−	0.992748i	$-0.538357\pi$
$659$	−4067.28	−0.240423	−0.120211	+	0.992748i	$0.538357\pi$
$660$	−3136.49	−0.184981
$661$	−700.954	−0.0412465	−0.0206233	−	0.999787i	$-0.506565\pi$
$661$	−700.954	−0.0412465	−0.0206233	+	0.999787i	$0.506565\pi$
$662$	−9307.40	−0.546438
$663$	2230.81	0.130675
$664$	1723.84	0.100750
$665$	25070.6	1.46195
$666$	−11374.3	−0.661777
$667$	3968.56	0.230379
$668$	17227.6	0.997837
$669$	−2644.94	−0.152854
$670$	−11634.3	−0.670852
$671$	11170.6	0.642677
$672$	3578.22	0.205406
$673$	24721.5	1.41597	0.707983	−	0.706229i	$-0.249605\pi$
$673$	24721.5	1.41597	0.707983	+	0.706229i	$0.249605\pi$
$674$	−2541.69	−0.145256
$675$	11291.1	0.643845
$676$	12563.0	0.714783
$677$	10400.6	0.590442	0.295221	−	0.955429i	$-0.404607\pi$
$677$	10400.6	0.590442	0.295221	+	0.955429i	$0.404607\pi$
$678$	−396.283	−0.0224472
$679$	7974.09	0.450688
$680$	−27730.4	−1.56384
$681$	−1584.41	−0.0891551
$682$	4385.75	0.246245
$683$	−18255.8	−1.02275	−0.511376	−	0.859357i	$-0.670864\pi$
$683$	−18255.8	−1.02275	−0.511376	+	0.859357i	$0.670864\pi$
$684$	12765.1	0.713578
$685$	375.006	0.0209171
$686$	5982.31	0.332953
$687$	4007.76	0.222570
$688$	0	0
$689$	−11102.6	−0.613897
$690$	800.326	0.0441564
$691$	30571.0	1.68304	0.841518	−	0.540229i	$-0.181663\pi$
$691$	30571.0	1.68304	0.841518	+	0.540229i	$0.181663\pi$
$692$	−11417.5	−0.627208
$693$	11594.8	0.635570
$694$	9668.49	0.528834
$695$	−54444.3	−2.97150
$696$	1704.35	0.0928208
$697$	39471.0	2.14501
$698$	−2996.68	−0.162502
$699$	937.866	0.0507487
$700$	26460.4	1.42872
$701$	9418.96	0.507488	0.253744	−	0.967271i	$-0.418338\pi$
$701$	9418.96	0.507488	0.253744	+	0.967271i	$0.418338\pi$
$702$	1183.85	0.0636487
$703$	30744.5	1.64943
$704$	−3484.63	−0.186551
$705$	−8804.12	−0.470330
$706$	1532.15	0.0816757
$707$	10868.1	0.578129
$708$	−4500.46	−0.238895
$709$	21799.2	1.15471	0.577353	−	0.816495i	$-0.304086\pi$
$709$	21799.2	1.15471	0.577353	+	0.816495i	$0.304086\pi$
$710$	−680.882	−0.0359902
$711$	−17787.1	−0.938209
$712$	−11822.4	−0.622278
$713$	7437.98	0.390680
$714$	−2280.05	−0.119508
$715$	−8027.20	−0.419861
$716$	1371.80	0.0716013
$717$	−4084.21	−0.212730
$718$	11837.9	0.615302
$719$	9463.10	0.490840	0.245420	−	0.969417i	$-0.421074\pi$
$719$	9463.10	0.490840	0.245420	+	0.969417i	$0.421074\pi$
$720$	18365.7	0.950625
$721$	−35900.7	−1.85439
$722$	−1824.43	−0.0940416
$723$	3228.06	0.166048
$724$	14743.9	0.756842
$725$	19346.0	0.991024
$726$	921.475	0.0471062
$727$	−7627.15	−0.389099	−0.194550	−	0.980893i	$-0.562325\pi$
$727$	−7627.15	−0.389099	−0.194550	+	0.980893i	$0.562325\pi$
$728$	5966.02	0.303730
$729$	−14495.4	−0.736444
$730$	9199.25	0.466410
$731$	0	0
$732$	3783.41	0.191037
$733$	−2047.25	−0.103161	−0.0515805	−	0.998669i	$-0.516426\pi$
$733$	−2047.25	−0.103161	−0.0515805	+	0.998669i	$0.516426\pi$
$734$	−3902.80	−0.196260
$735$	925.104	0.0464258
$736$	6455.28	0.323294
$737$	14543.9	0.726909
$738$	10228.6	0.510191
$739$	−2538.29	−0.126350	−0.0631749	−	0.998002i	$-0.520123\pi$
$739$	−2538.29	−0.126350	−0.0631749	+	0.998002i	$0.520123\pi$
$740$	53492.5	2.65733
$741$	−1562.59	−0.0774672
$742$	11347.6	0.561436
$743$	−18805.1	−0.928524	−0.464262	−	0.885698i	$-0.653680\pi$
$743$	−18805.1	−0.928524	−0.464262	+	0.885698i	$0.653680\pi$
$744$	3194.35	0.157406
$745$	−23182.2	−1.14004
$746$	2014.80	0.0988833
$747$	2904.04	0.142240
$748$	16120.1	0.787981
$749$	−1548.54	−0.0755438
$750$	1371.27	0.0667620
$751$	−33700.0	−1.63746	−0.818729	−	0.574181i	$-0.805321\pi$
$751$	−33700.0	−1.63746	−0.818729	+	0.574181i	$0.805321\pi$
$752$	−17789.2	−0.862639
$753$	−8110.20	−0.392499
$754$	2028.38	0.0979698
$755$	−20029.0	−0.965470
$756$	8042.01	0.386885
$757$	−28589.4	−1.37265	−0.686326	−	0.727294i	$-0.740778\pi$
$757$	−28589.4	−1.37265	−0.686326	+	0.727294i	$0.740778\pi$
$758$	1296.87	0.0621430
$759$	−1000.48	−0.0478461
$760$	19424.0	0.927083
$761$	−16653.2	−0.793268	−0.396634	−	0.917977i	$-0.629822\pi$
$761$	−16653.2	−0.793268	−0.396634	+	0.917977i	$0.629822\pi$
$762$	−1460.48	−0.0694324
$763$	3106.82	0.147411
$764$	−22585.6	−1.06953
$765$	−46715.7	−2.20785
$766$	−2581.36	−0.121760
$767$	−11518.0	−0.542231
$768$	302.914	0.0142324
$769$	19518.3	0.915279	0.457640	−	0.889138i	$-0.348695\pi$
$769$	19518.3	0.915279	0.457640	+	0.889138i	$0.348695\pi$
$770$	8204.38	0.383981
$771$	−1737.00	−0.0811370
$772$	−32230.9	−1.50261
$773$	633.350	0.0294696	0.0147348	−	0.999891i	$-0.495310\pi$
$773$	633.350	0.0294696	0.0147348	+	0.999891i	$0.495310\pi$
$774$	0	0
$775$	36258.8	1.68059
$776$	6178.11	0.285800
$777$	9458.27	0.436697
$778$	−12864.4	−0.592815
$779$	−27647.8	−1.27161
$780$	−2718.76	−0.124804
$781$	851.166	0.0389976
$782$	−4113.31	−0.188097
$783$	5879.76	0.268360
$784$	1869.22	0.0851502
$785$	43152.1	1.96199
$786$	3059.20	0.138827
$787$	22023.2	0.997511	0.498755	−	0.866743i	$-0.333791\pi$
$787$	22023.2	0.997511	0.498755	+	0.866743i	$0.333791\pi$
$788$	14818.4	0.669903
$789$	−1903.75	−0.0859003
$790$	−12586.0	−0.566821
$791$	−6889.63	−0.309693
$792$	8983.34	0.403042
$793$	9682.88	0.433605
$794$	−6396.84	−0.285914
$795$	−11120.5	−0.496104
$796$	−23933.6	−1.06571
$797$	−20062.0	−0.891635	−0.445817	−	0.895124i	$-0.647087\pi$
$797$	−20062.0	−0.891635	−0.445817	+	0.895124i	$0.647087\pi$
$798$	1597.08	0.0708472
$799$	45249.2	2.00351
$800$	31468.3	1.39072
$801$	−19916.4	−0.878541
$802$	1414.19	0.0622652
$803$	−11499.9	−0.505384
$804$	4925.93	0.216075
$805$	13914.2	0.609205
$806$	3801.65	0.166138
$807$	6051.49	0.263969
$808$	8420.31	0.366615
$809$	37497.8	1.62961	0.814804	−	0.579736i	$-0.196845\pi$
$809$	37497.8	1.62961	0.814804	+	0.579736i	$0.196845\pi$
$810$	−11499.3	−0.498820
$811$	−1705.17	−0.0738304	−0.0369152	−	0.999318i	$-0.511753\pi$
$811$	−1705.17	−0.0738304	−0.0369152	+	0.999318i	$0.511753\pi$
$812$	13779.0	0.595504
$813$	1460.21	0.0629910
$814$	10061.2	0.433223
$815$	−46950.5	−2.01792
$816$	4514.72	0.193685
$817$	0	0
$818$	14008.8	0.598785
$819$	10050.6	0.428811
$820$	−48104.7	−2.04865
$821$	−18361.5	−0.780535	−0.390267	−	0.920702i	$-0.627617\pi$
$821$	−18361.5	−0.780535	−0.390267	+	0.920702i	$0.627617\pi$
$822$	23.8891	0.00101366
$823$	14800.1	0.626851	0.313426	−	0.949613i	$-0.398523\pi$
$823$	14800.1	0.626851	0.313426	+	0.949613i	$0.398523\pi$
$824$	−27814.9	−1.17594
$825$	−4877.17	−0.205820
$826$	11772.2	0.495895
$827$	14542.4	0.611474	0.305737	−	0.952116i	$-0.401097\pi$
$827$	14542.4	0.611474	0.305737	+	0.952116i	$0.401097\pi$
$828$	7084.65	0.297353
$829$	32533.1	1.36299	0.681496	−	0.731822i	$-0.261330\pi$
$829$	32533.1	1.36299	0.681496	+	0.731822i	$0.261330\pi$
$830$	2054.87	0.0859346
$831$	−2931.60	−0.122378
$832$	−3020.54	−0.125864
$833$	−4754.61	−0.197764
$834$	−3468.29	−0.144001
$835$	44161.5	1.83027
$836$	−11291.5	−0.467134
$837$	11020.0	0.455087
$838$	−10853.2	−0.447396
$839$	−9279.94	−0.381858	−0.190929	−	0.981604i	$-0.561150\pi$
$839$	−9279.94	−0.381858	−0.190929	+	0.981604i	$0.561150\pi$
$840$	5975.63	0.245451
$841$	−14314.7	−0.586933
$842$	2159.05	0.0883680
$843$	4767.16	0.194768
$844$	12764.8	0.520595
$845$	32204.3	1.31108
$846$	11726.0	0.476534
$847$	16020.4	0.649903
$848$	−22469.5	−0.909911
$849$	3688.92	0.149121
$850$	−20051.7	−0.809136
$851$	17063.2	0.687330
$852$	288.285	0.0115921
$853$	−37303.3	−1.49735	−0.748675	−	0.662937i	$-0.769310\pi$
$853$	−37303.3	−1.49735	−0.748675	+	0.662937i	$0.769310\pi$
$854$	−9896.60	−0.396551
$855$	32722.4	1.30887
$856$	−1199.76	−0.0479055
$857$	−28599.1	−1.13994	−0.569969	−	0.821666i	$-0.693045\pi$
$857$	−28599.1	−1.13994	−0.569969	+	0.821666i	$0.693045\pi$
$858$	−511.360	−0.0203468
$859$	17588.9	0.698632	0.349316	−	0.937005i	$-0.386414\pi$
$859$	17588.9	0.698632	0.349316	+	0.937005i	$0.386414\pi$
$860$	0	0
$861$	−8505.63	−0.336668
$862$	13250.4	0.523564
$863$	16630.5	0.655978	0.327989	−	0.944681i	$-0.393629\pi$
$863$	16630.5	0.655978	0.327989	+	0.944681i	$0.393629\pi$
$864$	9564.06	0.376592
$865$	−29267.8	−1.15045
$866$	−1198.78	−0.0470397
$867$	−6029.59	−0.236189
$868$	25825.1	1.00986
$869$	15733.6	0.614186
$870$	2031.65	0.0791716
$871$	12606.9	0.490436
$872$	2407.08	0.0934793
$873$	10407.9	0.403497
$874$	2881.21	0.111508
$875$	23840.3	0.921085
$876$	−3894.95	−0.150226
$877$	−29568.5	−1.13849	−0.569247	−	0.822167i	$-0.692765\pi$
$877$	−29568.5	−1.13849	−0.569247	+	0.822167i	$0.692765\pi$
$878$	−16777.1	−0.644875
$879$	211.717	0.00812405
$880$	−16245.5	−0.622313
$881$	3239.15	0.123870	0.0619351	−	0.998080i	$-0.480273\pi$
$881$	3239.15	0.123870	0.0619351	+	0.998080i	$0.480273\pi$
$882$	−1232.12	−0.0470382
$883$	9200.23	0.350637	0.175318	−	0.984512i	$-0.443904\pi$
$883$	9200.23	0.350637	0.175318	+	0.984512i	$0.443904\pi$
$884$	13973.2	0.531640
$885$	−11536.6	−0.438189
$886$	10309.3	0.390913
$887$	−11534.9	−0.436643	−0.218322	−	0.975877i	$-0.570058\pi$
$887$	−11534.9	−0.436643	−0.218322	+	0.975877i	$0.570058\pi$
$888$	7328.01	0.276928
$889$	−25391.3	−0.957927
$890$	−14092.7	−0.530773
$891$	14375.2	0.540502
$892$	−16567.2	−0.621873
$893$	−31695.2	−1.18773
$894$	−1476.78	−0.0552471
$895$	3516.50	0.131334
$896$	28872.5	1.07652
$897$	−867.237	−0.0322811
$898$	−6676.68	−0.248111
$899$	18881.5	0.700482
$900$	34536.4	1.27912
$901$	57154.2	2.11330
$902$	−9047.79	−0.333990
$903$	0	0
$904$	−5337.90	−0.196389
$905$	37794.9	1.38823
$906$	−1275.91	−0.0467874
$907$	27277.5	0.998603	0.499302	−	0.866428i	$-0.333590\pi$
$907$	27277.5	0.998603	0.499302	+	0.866428i	$0.333590\pi$
$908$	−9924.32	−0.362720
$909$	14185.2	0.517593
$910$	7111.71	0.259067
$911$	676.309	0.0245962	0.0122981	−	0.999924i	$-0.496085\pi$
$911$	676.309	0.0245962	0.0122981	+	0.999924i	$0.496085\pi$
$912$	−3162.38	−0.114821
$913$	−2568.78	−0.0931154
$914$	−5523.88	−0.199905
$915$	9698.48	0.350406
$916$	25103.6	0.905508
$917$	53186.1	1.91533
$918$	−6094.23	−0.219106
$919$	1872.82	0.0672239	0.0336119	−	0.999435i	$-0.489299\pi$
$919$	1872.82	0.0672239	0.0336119	+	0.999435i	$0.489299\pi$
$920$	10780.3	0.386322
$921$	−4819.55	−0.172432
$922$	−6448.76	−0.230346
$923$	737.806	0.0263112
$924$	−3473.73	−0.123677
$925$	83179.8	2.95669
$926$	19683.5	0.698530
$927$	−46858.0	−1.66022
$928$	16386.9	0.579662
$929$	−7059.21	−0.249306	−0.124653	−	0.992200i	$-0.539782\pi$
$929$	−7059.21	−0.249306	−0.124653	+	0.992200i	$0.539782\pi$
$930$	3807.77	0.134260
$931$	3330.41	0.117239
$932$	5874.55	0.206467
$933$	6763.15	0.237316
$934$	−2776.17	−0.0972582
$935$	41322.6	1.44534
$936$	7786.92	0.271927
$937$	−33383.4	−1.16392	−0.581958	−	0.813219i	$-0.697713\pi$
$937$	−33383.4	−1.16392	−0.581958	+	0.813219i	$0.697713\pi$
$938$	−12885.2	−0.448525
$939$	6230.42	0.216530
$940$	−55146.7	−1.91350
$941$	24876.3	0.861791	0.430895	−	0.902402i	$-0.358198\pi$
$941$	24876.3	0.861791	0.430895	+	0.902402i	$0.358198\pi$
$942$	2748.93	0.0950797
$943$	−15344.5	−0.529891
$944$	−23310.2	−0.803690
$945$	20615.1	0.709638
$946$	0	0
$947$	−44261.5	−1.51880	−0.759400	−	0.650624i	$-0.774508\pi$
$947$	−44261.5	−1.51880	−0.759400	+	0.650624i	$0.774508\pi$
$948$	5328.89	0.182568
$949$	−9968.35	−0.340976
$950$	14045.4	0.479676
$951$	4400.45	0.150047
$952$	−30712.0	−1.04557
$953$	5408.14	0.183827	0.0919133	−	0.995767i	$-0.470702\pi$
$953$	5408.14	0.183827	0.0919133	+	0.995767i	$0.470702\pi$
$954$	14811.1	0.502648
$955$	−57896.5	−1.96177
$956$	−25582.4	−0.865476
$957$	−2539.75	−0.0857873
$958$	−11466.5	−0.386709
$959$	415.327	0.0139850
$960$	−3025.41	−0.101713
$961$	5597.27	0.187885
$962$	8721.20	0.292290
$963$	−2021.17	−0.0676337
$964$	20219.7	0.675553
$965$	−82621.5	−2.75614
$966$	886.379	0.0295225
$967$	−3500.75	−0.116418	−0.0582092	−	0.998304i	$-0.518539\pi$
$967$	−3500.75	−0.116418	−0.0582092	+	0.998304i	$0.518539\pi$
$968$	12412.2	0.412131
$969$	8043.94	0.266676
$970$	7364.52	0.243774
$971$	988.893	0.0326829	0.0163414	−	0.999866i	$-0.494798\pi$
$971$	988.893	0.0326829	0.0163414	+	0.999866i	$0.494798\pi$
$972$	15867.4	0.523607
$973$	−60298.3	−1.98672
$974$	8899.98	0.292786
$975$	−4227.62	−0.138864
$976$	19596.3	0.642685
$977$	43478.3	1.42374	0.711870	−	0.702311i	$-0.247848\pi$
$977$	43478.3	1.42374	0.711870	+	0.702311i	$0.247848\pi$
$978$	−2990.91	−0.0977900
$979$	17617.2	0.575125
$980$	5794.61	0.188880
$981$	4055.06	0.131976
$982$	−1661.35	−0.0539877
$983$	43582.2	1.41410	0.707048	−	0.707166i	$-0.250027\pi$
$983$	43582.2	1.41410	0.707048	+	0.707166i	$0.250027\pi$
$984$	−6589.93	−0.213495
$985$	37985.8	1.22876
$986$	−10441.7	−0.337254
$987$	−9750.76	−0.314458
$988$	−9787.66	−0.315169
$989$	0	0
$990$	10708.5	0.343775
$991$	−32709.0	−1.04847	−0.524236	−	0.851573i	$-0.675649\pi$
$991$	−32709.0	−1.04847	−0.524236	+	0.851573i	$0.675649\pi$
$992$	30712.8	0.982996
$993$	10101.8	0.322830
$994$	−754.092	−0.0240627
$995$	−61352.0	−1.95476
$996$	−870.031	−0.0276787
$997$	29341.0	0.932035	0.466017	−	0.884776i	$-0.345688\pi$
$997$	29341.0	0.932035	0.466017	+	0.884776i	$0.345688\pi$
$998$	−14390.8	−0.456445
$999$	25280.6	0.800642

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.j.1.29	yes	50
43.42	odd	2		1849.4.a.i.1.22	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.22	✓	50	43.42	odd	2
1849.4.a.j.1.29	yes	50	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q+1.02286 q^{2} -1.11016 q^{3} -6.95376 q^{4} -17.8254 q^{5} -1.13554 q^{6} -19.7420 q^{7} -15.2956 q^{8} -25.7675 q^{9} +O(q^{10})\)	\(q+1.02286 q^{2} -1.11016 q^{3} -6.95376 q^{4} -17.8254 q^{5} -1.13554 q^{6} -19.7420 q^{7} -15.2956 q^{8} -25.7675 q^{9} -18.2329 q^{10} +22.7928 q^{11} +7.71979 q^{12} +19.7572 q^{13} -20.1933 q^{14} +19.7891 q^{15} +39.9848 q^{16} -101.707 q^{17} -26.3566 q^{18} +71.2415 q^{19} +123.954 q^{20} +21.9168 q^{21} +23.3139 q^{22} +39.5390 q^{23} +16.9806 q^{24} +192.745 q^{25} +20.2089 q^{26} +58.5805 q^{27} +137.281 q^{28} +100.371 q^{29} +20.2414 q^{30} +188.118 q^{31} +163.264 q^{32} -25.3037 q^{33} -104.032 q^{34} +351.910 q^{35} +179.181 q^{36} +431.553 q^{37} +72.8700 q^{38} -21.9337 q^{39} +272.650 q^{40} -388.086 q^{41} +22.4178 q^{42} -158.496 q^{44} +459.317 q^{45} +40.4428 q^{46} -444.898 q^{47} -44.3896 q^{48} +46.7482 q^{49} +197.151 q^{50} +112.911 q^{51} -137.387 q^{52} -561.950 q^{53} +59.9196 q^{54} -406.292 q^{55} +301.966 q^{56} -79.0895 q^{57} +102.665 q^{58} -582.977 q^{59} -137.608 q^{60} +490.093 q^{61} +192.418 q^{62} +508.704 q^{63} -152.883 q^{64} -352.181 q^{65} -25.8821 q^{66} +638.092 q^{67} +707.245 q^{68} -43.8946 q^{69} +359.955 q^{70} +37.3436 q^{71} +394.130 q^{72} -504.541 q^{73} +441.418 q^{74} -213.978 q^{75} -495.396 q^{76} -449.977 q^{77} -22.4351 q^{78} +690.289 q^{79} -712.746 q^{80} +630.690 q^{81} -396.958 q^{82} -112.701 q^{83} -152.404 q^{84} +1812.97 q^{85} -111.428 q^{87} -348.630 q^{88} +772.926 q^{89} +469.817 q^{90} -390.048 q^{91} -274.945 q^{92} -208.841 q^{93} -455.068 q^{94} -1269.91 q^{95} -181.249 q^{96} -403.914 q^{97} +47.8168 q^{98} -587.315 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9}+O(q^{10}) \) 50 * q + 10 * q^2 + 2 * q^3 + 186 * q^4 - 8 * q^5 - 51 * q^6 + 6 * q^7 + 138 * q^8 + 360 * q^9	\( 50 q + 10 q^{2} + 2 q^{3} + 186 q^{4} - 8 q^{5} - 51 q^{6} + 6 q^{7} + 138 q^{8} + 360 q^{9} - 137 q^{10} - 252 q^{11} + 48 q^{12} - 192 q^{13} - 272 q^{14} - 314 q^{15} + 542 q^{16} - 236 q^{17} + 386 q^{18} - 12 q^{19} - 108 q^{20} - 408 q^{21} - 1235 q^{22} - 630 q^{23} - 613 q^{24} + 1098 q^{25} - 1493 q^{26} - 10 q^{27} + 242 q^{28} - 208 q^{29} + 48 q^{30} - 932 q^{31} + 1124 q^{32} - 254 q^{33} - 765 q^{34} - 1452 q^{35} + 747 q^{36} + 90 q^{37} - 1213 q^{38} + 1610 q^{39} - 1693 q^{40} - 1354 q^{41} + 16 q^{42} - 2704 q^{44} - 4508 q^{45} - 233 q^{46} - 3484 q^{47} + 376 q^{48} + 1324 q^{49} + 408 q^{50} - 4054 q^{51} - 2176 q^{52} - 726 q^{53} - 6497 q^{54} + 3288 q^{55} - 7097 q^{56} - 870 q^{57} + 275 q^{58} - 4370 q^{59} - 3891 q^{60} - 1172 q^{61} + 1546 q^{62} + 3686 q^{63} + 606 q^{64} - 2610 q^{65} - 4697 q^{66} - 344 q^{67} - 3221 q^{68} - 136 q^{69} + 1310 q^{70} - 162 q^{71} + 5814 q^{72} - 746 q^{73} - 4332 q^{74} - 236 q^{75} - 1338 q^{76} - 2024 q^{77} - 2782 q^{78} - 2656 q^{79} + 5713 q^{80} - 86 q^{81} + 4168 q^{82} - 3514 q^{83} - 4269 q^{84} + 7558 q^{85} - 10278 q^{87} - 11692 q^{88} - 2640 q^{89} - 8286 q^{90} + 5946 q^{91} - 4271 q^{92} + 2 q^{93} - 9062 q^{94} - 12140 q^{95} - 700 q^{96} - 3864 q^{97} + 2826 q^{98} - 8174 q^{99}+O(q^{100}) \) 50 * q + 10 * q^2 + 2 * q^3 + 186 * q^4 - 8 * q^5 - 51 * q^6 + 6 * q^7 + 138 * q^8 + 360 * q^9 - 137 * q^10 - 252 * q^11 + 48 * q^12 - 192 * q^13 - 272 * q^14 - 314 * q^15 + 542 * q^16 - 236 * q^17 + 386 * q^18 - 12 * q^19 - 108 * q^20 - 408 * q^21 - 1235 * q^22 - 630 * q^23 - 613 * q^24 + 1098 * q^25 - 1493 * q^26 - 10 * q^27 + 242 * q^28 - 208 * q^29 + 48 * q^30 - 932 * q^31 + 1124 * q^32 - 254 * q^33 - 765 * q^34 - 1452 * q^35 + 747 * q^36 + 90 * q^37 - 1213 * q^38 + 1610 * q^39 - 1693 * q^40 - 1354 * q^41 + 16 * q^42 - 2704 * q^44 - 4508 * q^45 - 233 * q^46 - 3484 * q^47 + 376 * q^48 + 1324 * q^49 + 408 * q^50 - 4054 * q^51 - 2176 * q^52 - 726 * q^53 - 6497 * q^54 + 3288 * q^55 - 7097 * q^56 - 870 * q^57 + 275 * q^58 - 4370 * q^59 - 3891 * q^60 - 1172 * q^61 + 1546 * q^62 + 3686 * q^63 + 606 * q^64 - 2610 * q^65 - 4697 * q^66 - 344 * q^67 - 3221 * q^68 - 136 * q^69 + 1310 * q^70 - 162 * q^71 + 5814 * q^72 - 746 * q^73 - 4332 * q^74 - 236 * q^75 - 1338 * q^76 - 2024 * q^77 - 2782 * q^78 - 2656 * q^79 + 5713 * q^80 - 86 * q^81 + 4168 * q^82 - 3514 * q^83 - 4269 * q^84 + 7558 * q^85 - 10278 * q^87 - 11692 * q^88 - 2640 * q^89 - 8286 * q^90 + 5946 * q^91 - 4271 * q^92 + 2 * q^93 - 9062 * q^94 - 12140 * q^95 - 700 * q^96 - 3864 * q^97 + 2826 * q^98 - 8174 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more